• In this lab you will take data from a video and attemptto verify the Law of Conservation of Momentum. Additionally, youwill take into account the uncertainty of (most of) themeasurements.
  
  There is no such thing as a perfect measurement. All measurementshave some amount of error. Some of that error comes from mistakesmade while taking the measurement; by slightly misusing theequipment (for example, not perfectly lining up a ruler) or bymisreading the equipment. As such, it is common to state theuncertainty of a measurement. This is done by using the plus/minussymbol; ±. The number following this symbol is the uncertainty. Forexample, the measurement \"5.2 m ± 0.2 m\" has an uncertainty of 0.2m. Overall, this means: \"We believe the value is 5.2 m, weacknowledge we are probably slightly incorrect, but we aresupremely confident that the actual value lies between 5 m &5.4 m.\"
  
  In this lab you will be asked to estimate one uncertainty yourself.The rest will either be given to you or you will calculate themusing the formulas provided.
  

• Video & data table

  The video below shows a dart being fired into a cart (thatis initially at rest). The beginning sequence was filmed at 240frames per second and it took 10 frames for the dart to travel 31cm. You can use this data to determine the dart's momentum prior toimpact. For this calculation, we will assume that the values thatwere just stated (240 frames per second, 10 frames, & 31 cm)are all exact. Fill in the PRE-COLLISION DATA TABLE.
  
  After the collision, the frame rate is 60 frames per second. Thevideo shows the cart (with the dart embedded in the foam block)moving forward. The frame counter has restarted so that the \"X\" onthe cart is at 0 cm at frame zero. By advancing the video,determine at which frame the \"X\" reaches the 10 cm line and enterthat data in the POST-COLLISION DATA TABLE. You'll also need toidentify the frame uncertainty: How many frames could your data beoff by? You can assume that the 10 cm & 60 frames per secondare exact values.
  
  Other data that you need is stated below the video as well as theformulas for calculating speed uncertainty & momentumuncertainty. NOTE: LEAVE THE MASS IN GRAMS.
  
  Mass of dart = 6.1 g ± 0.1 g
  Mass of cart = 262.5 g ± 0.1 g
  Total mass = 268.6 g ± 0.2 g
  speed uncertainty =

  speed × frameuncertainty

  # of frames

  momentum uncertainty = mass uncertainty × speed +mass × speed uncertainty

  
  Frame uncertainty =  
  

  PRE-COLLISION DATA TABLE	# of frames needed for dart to travel 31 cm	Δt (s)	vdart,i (m/s)	vdart,i uncertainty (m/s)	pdart,i (g·m/s)	pdart,i uncertainty (g·m/s)
  Dart	10			0

  POST-COLLISION DATA TABLE	# of frames needed for cart to travel 10 cm	Δt (s)	vf (m/s)	vf uncertainty (m/s)	pf (g·m/s)	pf uncertainty (g·m/s)
  Cart + dart

• Momentum range

  Using the values you calculated for momentum & momentumuncertainty, state the range for the pre-collision momentum &the post-collision momentum in the table below.

  	minimum momentum (g·m/s)	maximum momentum (g·m/s)
  Dart (pre-collision)
  Cart + dart (post-collision)

  
  Ideally, these ranges will overlap. The claim is thatmomentum is conserved. Within experimental error, you have shownthis is true for this collision.

• Post-collision velocity

  By applying the Law of Conservation of Momentum to thissituation, derive a formula for v_f. Your answer shouldbe symbolic (no data). Use m for the dart mass, M for the cartmass, and v for the initial velocity of the dart.
  v_f =
  
  Using the formula you just derived, determine whatv_f will be for the following three cases:
  
  M = 0 (cart mass is zero); v_f =
  M = m (cart & dart have same mass); v_f =
  M >> m (the cart mass is much greater than the dart mass);v_f =
  
  Using your data and the formula you derived (at the top ofthis section), calculate v_f.
  v_f =  
  
  Now determine the uncertainty of this value using thefollowing formula: v_f uncertainty =

  dart massuncertainty × dart speed

  totalmass of cart & dart

  
  v_f uncertainty =  
  
  You have now determined v_f two different ways.In the POST-COLLISION DATA TABLE you determined it usingdistance/time. Just above, you determined it using conservation ofmomentum. Using the values & uncertainties you calculated foreach, fill in the table below. As before, ideally the ranges willoverlap.

  	minimum vf (m/s)	maximum vf (m/s)
  Determined using distance/time
  Determined using conservation of momentum